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SPLITTING FUNCTIONS AT NLO Germán F. R. Sborlini PhD. Student - Departmento de Física, FCEyN, UBA Advisor: Daniel de Florian PASI 2012 March 7, 2012 CONTENTS • Introduction • Collinear limits and definitions • Brief example • Conclusions and perspectives INTRODUCTION • Higher order corrections to QCD processes involves dealing with divergences • There are two kinds: • High-energy or UV singularities 1. Origin: Unknown high-energy physics. 2. Solution: Renormalization. • Low-energy or IR singularities 1. Origin: Emission of collinear or soft particles. 2. Solution: Some theorems (KLN) guarantee that they can be cancelled in the final physical result (if we are computing IR safe observables…). For instance, the cancellation can be implemented through the subtraction method. (See: Muta, Foundations of Quantum Chromodynamics) INTRODUCTION • Focusing on IR singularities • Related with low-energy and collinear configurations Soft configuration Collinear configuration (Note that these quantities appear in denominators of scattering amplitudes) • Associated with degenerate states (the experiments are not able to distinguish two particles which are very close neither they can detect low-energy particles) + Virtual corrections = Real corrections Finite result COLLINEAR LIMITS AND DEFINITIONS • Color decomposition: We subtract the color structure from the QCD amplitude, so we can introduce the color-ordered amplitudes (which contains kinematical information). • Tree-level decomposition (only external gluons) • One-loop decomposition (only external gluons) (More details about color decomposition: Dixon, arXiv:hep-ph/9601359v2) • Properties of color-ordered amplitudes • They are singular when two adjacent legs become collinear. • Subleading-color amplitudes are related to leading ones and have a similar collinear behaviour. (For more details see: Kosower and Uwer, arXiv:hep-ph/9903515) COLLINEAR LIMITS AND DEFINITIONS • Collinear factorization of color-ordered amplitudes: When we are in a collinear configuration, we can introduce universal functions (i.e. process-independent) which describe the collinear behaviour. These are the splitting functions. • Tree-level factorization Splitting function Reduced amplitude (n-1 external legs) (For more details see: Kosower and Uwer, arXiv:hep-ph/9903515) COLLINEAR LIMITS AND DEFINITIONS • Collinear factorization of color-ordered amplitudes • One-loop-level factorization (For more details see: Kosower and Uwer, arXiv:hep-ph/9903515) COLLINEAR LIMITS AND DEFINITIONS • Collinear factorization of color-ordered amplitudes • • At one-loop level it is possible to get an explicit formula to compute the splitting function Some remarks: • It is important to sum over the physical polarizations of the intermediate states. • If we sum over polarization of gluons we get (Again, see: Dixon, arXiv:hep-ph/9601359v2) where q is a reference null-vector (not collinear with a or b). This is similar to the light-cone gauge propagator. (For more details see: Kosower and Uwer, arXiv:hep-ph/9903515) COLLINEAR LIMITS AND DEFINITIONS • Collinear factorization in color space: It is possible to introduce splitting amplitudes working in the color space. They include color information (and are equal to splitting functions when there is only one colour structure). • It is useful to introduce some kinematical variables to parametrize collinear momenta. Particles 1 and 2 become collinear. n and p are null-vectors (p is the collinear direction) zi are the momentum fractions • Null-vector Matrix elements have an specific behaviour in the collinear limit: (It can be obtained rescaling the collinear momenta.) (For more details see: Catani, de Florian and Rodrigo, arXiv:1112.4405v1) COLLINEAR LIMITS AND DEFINITIONS • Collinear factorization in color space: We introduce the splitting matrices (which are the analogous of splitting functions but in color+spin space) • Tree-level factorization Splliting matrix at LO • One-loop level factorization Splliting matrix at NLO • Some remarks: • It is important to note that these expressions are valid up to order O(sqrt(s12)). • Sp includes color and spin information. • At LO we have the relation (For more details see: Catani, de Florian and Rodrigo, arXiv:1112.4405v1) COLLINEAR LIMITS AND DEFINITIONS • Collinear factorization in color space • General structure of one-loop splitting matrices One-loop splitting matrix • Finite contribution Singular contribution Tree-level splitting matrix More details: • SpH contains only rational functions of the momenta and only depends on collinear particles. • IC contains trascendental functions and can depend of non-collinear particles (through colour correlations). This contribution introduces a violation of strict collinear factorization. (For more details see: Catani, de Florian and Rodrigo, arXiv:1112.4405v1) COLLINEAR LIMITS AND DEFINITIONS • Relation between splitting functions and Altarelli-Parisi kernels: • Altarelli-Parisi kernels are related to the collinear behaviour of squared matrix elements. They also control the evolution of PDF’s and FF’s (through DGLAP equations). • LO contribution • NLO contribution • NOTE: Here we are choosing not to include the coupling constant inside the definition of splitting matrices. Also, to get AP kernels, we sum over final-state colors and spins, and average over colors and spins of the parent parton. (For more details see: Catani, de Florian and Rodrigo, arXiv:hep-ph/0312067) BRIEF EXAMPLE • g->gg splitting function Tree-level contribution One-loop contribution (It only contains the leading color contribution) • To compute the splitting function we have to calculate the leading-color amplitude in the collinear limit. There are some useful properties (and definitions): (For more details see: Kosower and Uwer, arXiv:hep-ph/9903515) BRIEF EXAMPLE • g->gg splitting function = NLO corrections proportional to LO contribution (For more details see: Kosower and Uwer, arXiv:hep-ph/9903515) CONCLUSIONS AND PERSPECTIVES • Splitting functions describe collinear behaviour of scattering amplitudes. • They are process-independent (at least at NLO…) and allow to compute Altarelli-Parisi kernels at higher orders. What are we trying to do? • Compute splitting amplitudes for multiple-collinear configurations at NLO (for example, q -> q Qb Q) • Use more efficient computation methods • Obtain results that are valid to all orders in the regularization parameter İ(remember the advantages of using dimensional regularization). This is one of our biggest challenges, VLQFHWKHUHVRPHLQWHJUDOVZKLFKDUHRQO\NQRZQWR2İ 0) (so, we must also study new methods to compute them properly). REFERENCES Kosower, D. and Uwer, P. – One-Loop Splitting Amplitudes in Gauge Theory (arXiv:hep-ph/9903515) Catani, S., de Florian, D. and Rodrigo, G. – Space-like (vs. timelike) collinear limits in QCD: is factorization violated? (arXiv:1112.4405) Catani, S., de Florian, D. and Rodrigo, G. – The triple collinear limit of one-loop QCD amplitudes (arXiv:hep-ph/0312067) Bern, Z., Dixon, L. and Kosower, D. – Two-Loop g->gg Splitting Amplitudes in QCD (arXiv:hep-ph/0404293v2)